 2.4.1: Write gcd(308, 165) as a linear combination of 308 and 165
 2.4.2: Write gcd(2420, 70) as a linear combination of 2420 and 70
 2.4.3: Write gcd(735, 90) as a linear combination of 735 and 90
 2.4.4: Write gcd(8370, 465) as a linear combination of 8370 and 465.
 2.4.5: Write gcd(1326, 252) as a linear combination of 1326 and 252.
 2.4.6: Write gcd(1018215, 2695) as a linear combination of 1018215 and 2695
 2.4.7: For Exercises 712, test whether n is prime and, if not, find its de...
 2.4.8: For Exercises 712, test whether n is prime and, if not, find its de...
 2.4.9: For Exercises 712, test whether n is prime and, if not, find its de...
 2.4.10: For Exercises 712, test whether n is prime and, if not, find its de...
 2.4.11: For Exercises 712, test whether n is prime and, if not, find its de...
 2.4.12: . The least common multiple of two positive integers a and b, lcm(a...
 2.4.13: Prove that for any positive integers a and b, a # b = gcd(a, b) # l...
 2.4.14: Find gcd(2420, 70) by unique factorization into products of primes
 2.4.15: Find gcd(735, 90) by unique factorization into products of primes.
 2.4.16: . The least common multiple of two positive integers a and b, lcm(a...
 2.4.17: Find gcd(1326, 252) by unique factorization into products of primes.
 2.4.18: Find gcd(1018215, 2695) by unique factorization into products of pr...
 2.4.19: The least common multiple of two positive integers a and b, lcm(a, ...
 2.4.20: Prove that for any positive integers a and b, a # b = gcd(a, b) # l...
 2.4.21: For Exercises 2124, find the gcd and lcm of the two numbers given.a...
 2.4.22: For Exercises 2124, find the gcd and lcm of the two numbers given.
 2.4.23: For Exercises 2124, find the gcd and lcm of the two numbers given.
 2.4.24: For Exercises 2124, find the gcd and lcm of the two numbers given.
 2.4.25: Prove that for any positive integers a and b, gcd(a, b) = gcd(a, a ...
 2.4.26: Prove that gcd(n, n + 1) = 1 for all positive integers n.
 2.4.27: Find an example where n 0 ab but n  a and n  b. Does this violate...
 2.4.28: The division of a full circle into 360 probably dates back to the e...
 2.4.29: Prove that there exist three consecutive odd positive integers that...
 2.4.30: Prove that for any positive integer n, there exist n consecutive co...
 2.4.31: For Exercises 3134, find (n) together with the numbers that give th...
 2.4.32: For Exercises 3134, find (n) together with the numbers that give th...
 2.4.33: For Exercises 3134, find (n) together with the numbers that give th...
 2.4.34: For Exercises 3134, find (n) together with the numbers that give th...
 2.4.35: By Practice 16, if p is prime then (p) = p 1. Prove that this is an...
 2.4.36: For any prime number p and any positive integer k, (pk ) = pk1 (p)....
 2.4.37: Compute (24 ) and state the numbers being counted. (Hint: See Exerc...
 2.4.38: Compute (33 ) and state the numbers being counted. (Hint: See Exerc...
 2.4.39: n = 117612 = 22 # 35 # 112
 2.4.40: n = 233206 = 2 # 17 # 193
 2.4.41: n = 1795625 = 54 # 132 # 17
 2.4.42: n = 1,690,541,699 = 74 # 113 # 232
 2.4.43: If p and q are prime numbers with p q, then (pq) = (p)(q). Although...
 2.4.44: Prove that if r and s are relatively prime, then (rs) = (r) (s).
 2.4.45: Prove that for n and m positive integers, (nm) = nm1 (n)
 2.4.46: Except for n = 2, all the values of (n) in Example 34 are even numb...
 2.4.47: A particular class of prime numbers is known as Mersenne primes, na...
 2.4.48: Goldbachs conjecture states that every even integer greater than 2 ...
 2.4.49: A perfect number is a positive integer n that equals the sum of all...
 2.4.50: a. Prove that 496 is a perfect number (see Exercise 49) by writing ...
 2.4.51: An algorithm exists to find all prime numbers less than some given ...
 2.4.52: a. Compute the square of 11. b. Compute the square of 111. c. Prove...
 2.4.53: Sudoku puzzles are popular numberbased puzzles. A game consists of...
Solutions for Chapter 2.4: Number Theory
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 2.4: Number Theory
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.4: Number Theory includes 53 full stepbystep solutions. Since 53 problems in chapter 2.4: Number Theory have been answered, more than 21624 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Solvable system Ax = b.
The right side b is in the column space of A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.